FEYNHELPERS: CONNECTING FEYNCALC TO FIRE AND PACKAGE-X

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1 FEYNHELPERS: CONNECTING FEYNCALC TO FIRE AND PACKAGE-X Vladyslav Shtabovenko Technische Universität München Instituto de Física Corpuscular, Valencia Physik-Department T30f Physik-Department T30f (TUM) FeynHelpers 1 / 27

2 OUTLINE 1 FEYNCALC: WHAT IS IT AND WHAT IS IT GOOD FOR? 2 FEYNCALC AND (MULTI-)LOOP INTEGRALS: STRENGTHS AND WEAKNESSES 3 FEYNHELPERS: GOING BEYOND BOUNDARIES 4 SUMMARY AND OUTLOOK: WHERE WE ARE NOW AND WHERE WE ARE GOING Physik-Department T30f (TUM) FeynHelpers 2 / 27

3 A generic perturbative QFT calculation may involve many different steps Feynman diagrams Feynman rules from L Diagram generation Amplitudes... Dirac algebra Simplification of γ-matrix chains Dirac traces SPVAT form Fierz identities... Loop integrals Tensor reduction Partial fractioning Mapping of topologies IBP-Reduction Numerics We can automatize each step separately using standalone packages (e. g. FEYNARTS, LOOP- TOOLS [Hahn & Perez-Victoria, 1999], FEYNRULES [Christensen & Duhr, 2008], QGRAF [Nogueira, 1993], TRACER [Jamin & Lautenbacher, 1993], FORMTRACER [Cyrol et al., 2016], FORCER [Ruijl et al., 2017], PY- SECDEC [Borowka et al., ],... ) and self-written codes. Or we can employ all-in-one packages that handle most of these steps in one framework. Physik-Department T30f (TUM) FeynHelpers 3 / 27

4 Two big categories of all-in-one packages Fully-automatic (FORMCALC [Hahn & Perez-Victoria, 1999], GOSAM [Cullen et al., 2014], GRACE [Belanger et al., 2006], DIANA [Tentyukov & Fleischer, 2000], FDC [Wang, 2004],... ) Semi-automatic (FEYNCALC [Mertig et al., 1991, Shtabovenko et al., 2016], HEPMATH [Wiebusch, 2014], PACKAGE-X [Patel, 2015],... ) Fully-automatic tools Blackbox: Require only minimal user input and provide a small set of options. The code takes care of the rest. Semi-automatic tools Toolbox: Combine different tools with many options to get the computation done in your way. Easy to use Foolproof Constantly good performance Saves your time Limited number of templated calculations Difficult to extend/modify for your needs Not easy to obtain intermediate results You must know what you are doing Easy to make mistakes The performance depends on your skills Writing codes may take quite some time Very broad range of applications Extendable with user-defined objectsb Intermediate results at each step Physik-Department T30f (TUM) FeynHelpers 4 / 27

5 FEYNCALC Open source (GPLv3) MATHEMATICA package for symbolic semi-automatic evaluation of Feynman diagrams and algebraic expressions in QFT. Features Suitable for evaluating both single expressions and full Feynman diagrams. The calculation can be organized in many different ways (flexibility) Extensive typesetting for better readability Lorentz index contractions, SU(N) algebra, Dirac algebra, etc. Passarino-Veltman reduction of one-loop amplitudes to standard scalar integrals Basic support for manipulating multi-loop integrals General tools for non-commutative algebra BUT: Essentially only algebraic manipulations, everything else requires extra tools. Physik-Department T30f (TUM) FeynHelpers 5 / 27

6 FEYNCALC developer team Rolf Mertig (GluonVision GmbH): original author of the package, first release 1991 Frederik Orellana (Technical University of Denmark): joined 1997 VS (TUM, soon Zhejiang University): joined 2014 Recent developments (since 2014) Large parts of the code improved or rewritten from scratch. Public source code repository on GITHUB: Online documentation Ships with many sample calculations Extensive unit testing framework New and improved functions for loop calculations. Big emphasis on using FEYNCALC for Effective Field Theory (EFT) calculations. Original motivation for FEYNHELPERS: Matching calculations in relativistic EFTs Upcoming FEYNCALC 9.3 and FEYNONIUM: Matching calculations in nonrelativistic EFTs (in particular NRQED/NRQCD [Caswell & Lepage, 1986, Bodwin et al., 1995], pnrqed/pnrqcd [Pineda & Soto, 1998b, Pineda & Soto, 1998a, Brambilla et al., 2000] ) Physik-Department T30f (TUM) FeynHelpers 6 / 27

7 When is FEYNCALC useful? Small or medium-sized calculations, too specific for fully automatic packages FEYNCALC as a calculator for QFT expressions Cross-check results from other people Extensive manipulations on the level of the amplitudes Limitations of FEYNCALC? Written entirely in WOLFRAM language, cannot be used without MATHEMATICA Inherits MATHEMATICA s performance problems with large number of terms Not really suited for large and complex calculations Much slower than FORM Why not combine FEYNCALC/MATHEMATICA with FORM? Thomas Hahn already had a similar idea many years ago. FORMCALC is much faster than FEYNCALC, but also less flexible Performance-wise it is not so clever to constantly pass very large expressions between MATHEMATICA and FORM However, that would be necessary(?) to preserve the flexibility of FEYNCALC FEYNCALCFORMLINK employs FORM for index contractions and Dirac traces. FORMTRACER is a recent package that provides access to FORM from MATHEMATICA Physik-Department T30f (TUM) FeynHelpers 7 / 27

8 Most used functions for loop calculations ApartFF: Partial fractioning for 1-loop and multi-loop integrals FDS: Shifts in loop momenta for 1-loop and multi-loop integrals TID: Tensor reduction for 1-loop integrals ToPaVe: Converts scalar 1-loop integrals to Passarino Veltman scalar functions PaVeReduce: Reduction of Passarino Veltman coefficient functions to scalar functions FCMultiLoopTID: Tensor reduction for multi-loop integrals Less known functions FCLoopBasisIncompleteQ FCLoopBasisOverDeterminedQ FCLoopBasisFindCompletion FCLoopIBPReducableQ Physik-Department T30f (TUM) FeynHelpers 8 / 27

9 Partial fractioning Scalar loop integrals can be often simplified even further by using partial fractioning. Well known identities (implemented in SPC and Apart2) are q p = 1 2 [(q + p)2 + m 2 2 (q 2 + m 2 1) p 2 m m 2 1], 1 (q 2 m 2 1 )(q2 m 2 2 ) = 1 ( 1 m 2 1 m2 2 q 2 m 2 1 But: Many decompositions, e.g. d D 1 q q 2 (q p) 2 (q + p) = 1 2 p 2 ( d D q 1 q 2 m 2 2 ). 1 q 2 (q p) 1 2 (q p) 2 (q + p) 2 require more sophisticated algorithms. New in FEYNCALC 9: ApartFF introduces partial fractioning algorithm from [Feng, 2012] Compared to the reference MATHEMATICA implementation ( it is fully integrated into FEYNCALC ), In [1]:= ApartFF[FAD[{q}, {q p}, {q + p }], {q}] Out[2]:= 1 p 2 q 2.(q p) 1 2 p 2 q 2.(q 2p) 2 Physik-Department T30f (TUM) FeynHelpers 9 / 27

10 Tensor reduction 1-loop tensor reduction is done via Passarino-Veltman technique: TID TID has received many improvements in FEYNCALC 9 and above Default mode: Reduce each tensor integral to PaVe scalar functions (A 0, B 0, C 0, D 0 ) In [1]:= FCI[GAD[µ].(m + GSD[q]).GAD[µ] FAD[{q, m}]] Out[1]:= γµ.(m + γ q).γ µ (q 2 m 2 ).(q p) 2 In [2]:= TID[%, p + q], q ]// ToPaVe[#, q]& iπ 2 (D 2)A 0 (m 2) γ p Out[2]= 2p 2 iπ 2 B 0 (p 2, 0, m 2) ( ) Dm 2 γ p 2Dmp 2 + Dp 2 γ p 2m 2 γ p 2p 2 γ p 2p 2 Physik-Department T30f (TUM) FeynHelpers 10 / 27

11 Tensor reduction Zero Gram determinants? Detected automatically, reductions switches to Passarino Veltman coefficient functions (e.g. B 1, B 00, C 222 etc.) d D q γ µ (m + /q)γ µ Consider (2π) D (q 2 m 2 )(q p) with 2 p2 = 0 In [1]:= SPD[p, p] = 0; TID[GAD[µ].(m + GSD[q]).GAD[µ] FAD[{q, m}, p + q], q]; Out[2]:= iπ 2 B 0 ( 0, 0, m 2) (Dm Dγ p + 2γ p) iπ 2 (D 2)γ pb 1 ( 0, 0, m 2) Useful options: UsePaVeBasis: Enforces reduction into coefficient functions for any kinematics. GenPaVe: Allows define PaVe functions in a different way (standard is the LOOPTOOLS convention) Isolate: Kinematic coefficients in front of the loop inetgrals will be abbreviated. Use FRH to recover the original form. Physik-Department T30f (TUM) FeynHelpers 11 / 27

12 TENSOR REDUCTION How about multi-loop tensor reduction? In general, not very useful above 1-loop, many scalar products in the denominators can t be cancelled against propagators in the numerators. Still practical for loop momenta contracted with Dirac matrices and Levi-Civita tensors. FEYNCALC 9 features FCMultiLoopTID: uses the same PaVe algorithm as for 1-loop. currently no proper way to handle zero Gram determinants. In [1]:= FCI[FVD[q1, µ] FVD[q2, ν] FAD[q1, q2, {q1 p1}, {q2 p1}, {q1 q2}]] q1 µ q2 ν Out[1]:= q1 2.q2 2.(q1 p1) 2.(q2 p1) 2.(q1 q2) 2 In [2]:= FCMultiLoopTID[%, {q1, q2}] Out[2]:= Dp1 µ p1 ν p1 2 g µν 4(D 1)q2 2.q1 2.(q2 p1) 2.(q1 q2) 2.(q1 p1) 2 p1 2 g µν p1 µ p1 ν 2(D 1)p1 2 q2 2.q1 2.(q2 p1) 2.(q1 p1) + p1 2 g µν p1 µ p1 ν 2 (D 1)p1 2 q2 2.q1 2.(q1 q2) 2.(q1 p1) 2 Dp1 µ p1 ν p1 2 g µν 2(D 1)p1 4 q1 2.(q2 p1) 2.(q1 q2) 2 Physik-Department T30f (TUM) FeynHelpers 12 / 27

13 EXTRACTION OF LOOP INTEGRALS To evaluate the loop integrals outside of FEYNCALC, we need to extract all the unique integrals from the given expression New in FeynCalc 9: FCLoopIsolate In [1]:= gse = FCI[FAD[q, p + q] MTD[Lor3, Lor4] (FVD[ p q, Lor5] MTD[Lor1, Lor3] + FVD[2 p q, Lor3] MTD[Lor1, Lor5] + FVD[ p + 2 q, Lor1] MTD[Lor3, Lor5]) (FVD[p + q, Lor6] MTD[Lor2, Lor4] + FVD[ 2 p + q, Lor4] MTD[Lor2, Lor6] + FVD[p 2 q, Lor2] MTD[Lor4, Lor6]) MTD[Lor5, Lor6]] ( g Lor3Lor4 g Lor5Lor6 g Lor3Lor5 (2q p) Lor1 + g Lor1Lor5 (2p q) Lor3 + g Lor1Lor3 ( p q) Lor5) Out[1]:= q 2.(q p) (g 2 Lor4Lor6 (p 2q) Lor2 + g Lor2Lor6 (q 2p) Lor4 + g Lor2Lor4 (p + q) Lor6) In [2]:= FCLoopIsolate[Contract[gse], { {q}, Head > loop] // Cases2[#, loop] & ( ) ( ) ( ) 1 q Lor1 q Lor2 Out[2]:= loop, loop, loop, q 2.(q p) 2 q 2.(q p) 2 q 2.(q p) 2 ( ) q Lor1 q Lor2 ( ) ( )} pq q 2 loop, loop, loop q 2.(q p) 2 q 2.(q p) 2 q 2.(q p) 2 Furthermore: FCLoopSplit to separate different types of loop integrals (free of loops, scalar integrals with and without scalar products in the numerators, tensor integrals) FCLoopExtract for combined application of FCLoopIsolate and FCLoopSplit Physik-Department T30f (TUM) FeynHelpers 13 / 27

14 TOOLS FOR IBP-REDUCTION Reduction of scalar loop integrals using integration-by-parts (IBP) identities [Chetyrkin & Tkachov, 1981] is a standard technique in modern loop calculations. Many publicly available IBP-packages on the market: FIRE [Smirnov & Smirnov, 2013], LITERED [Lee, 2012], REDUZE [Studerus, 2009], AIR [Anastasiou & Lazopoulos, 2004],... Expected input: loop integrals with propagators that form a basis. What about integrals with an incomplete or overdetermined basis? FCLoopBasisIncompleteQ detects integrals that require a basis completion FCLoopBasisFindCompletion gives a list of propagators (with zero exponents) required to complete the basis FCLoopBasisOverdeterminedQ checks if the propagators are linearly dependent. Such integrals can be decomposed further using ApartFF. In [1]:= FCI[FAD[{q1, m, 2}, {q1 + q3, m}, {q2 q3}, q2]] Out[1]:= 1 ( q1 2 m 2). ( q1 2 m 2). ((q1 + q3) 2 m 2 ).(q2 q3) 2.q2 2 In [2]:= FCLoopBasisIncompleteQ[%, {q1, q2, q3}] Out[2]:= True In [3]:= FCLoopBasisFindCompletion[%%, {q1, q2, q3}][[2]] Out[3]:= {(q1 q2), (q1 q3)} Physik-Department T30f (TUM) FeynHelpers 14 / 27

15 Motivation The field of automatic calculations appears to be a very competitive environment. Some groups do not share their codes at all Others make them available to collaborators only. People behind similar software regarded as competitors. It is more efficient to combine useful tools together than to compete. Useful tools to be used with FEYNCALC for the evaluation of 1-loop integrals: FIRE [Smirnov, 2015] PACKAGE X [Patel, 2015] Challenges: Need to convert between the conventions used in each package and avoid variable shadowing. Solution: FEYNHELPERS [Shtabovenko, 2016] seamlessly integrates both tools into FeynCalc. FORM FeynArts FormLink FeynHelpers FeynCalc FeynHelpers FIRE Package-X Physik-Department T30f (TUM) FeynHelpers 15 / 27

16 Tensor reduction a la Passarino Veltman Very old technique for dealing with tensor 1-loop integrals [Passarino & Veltman, 1979] Still widely used in many loop calculations. Main idea: convert all the tensor integrals into scalar ones (Passarino Veltman coefficient functions) Evaluation of any 1-loop integral can be reduced to the evaluation of the resulting coefficient functions A lot of tools for numerical evaluation: FF [van Oldenborgh, 1991], LOOPTOOLS [Hahn & Perez-Victoria, 1999], QCDLOOP [Carrazza et al., 2016], ONELOOP [van Hameren, 2011], GOLEM95C [Cullen et al., 2011], PJFRY [Fleischer & Riemann, 2011], COLLIER [Denner et al., 2017],... Where to get analytic results for singular kinematics or zero Gram determinants? Often needed for renormalization, EFTs,... Most of the results can be found somewhere in the literature. PACKAGE-X Recent [Patel, 2015] MATHEMATICA package for semi-automatic 1-loop calculations (closed-source freeware) Unique feature: Library of analytic expressions for Passarino Veltman functions with up to 4 legs and almost arbitrary kinematics. Can also extract UV- and IR-parts and expand coefficient functions in their arguments. Someone indeed has collected all those results from the literature! Physik-Department T30f (TUM) FeynHelpers 16 / 27

17 Interface to PACKAGE-X Main function: PaXEvaluate Works: on scalar 1-loop integrals (unit numerators) and Passarino Veltman coefficient functions A, B, C and D Takes two arguments (plus options): input expression, loop momentum. Use PaXEvaluateUV(PaXEvaluateIR) to get the UV(IR)-divergent part of the result PaXEvaluateUVIRSplit returns the full result with the explicit distinction between ɛ UV and ɛ IR. All four functions share the same set of options Physik-Department T30f (TUM) FeynHelpers 17 / 27

18 Let us compute In[1]:= Out[1]= d D q (2π) D 1 q 2 m 2 int=paxevaluate[fad[{q,m}],q,paximplicitprefactor 1/(2Pi)^D] im 2 2 im ( log ( µ 2 ) ) +γ 1 log(4π) 16π 2 ε m 2 16 π 2 Make the result look more compact ( 1/ɛ γ E + log(4π)) using FCHideEpsilon In[2]:= Out[2]= int//fchideepsilon i m2 im 16π ( log ( µ 2 m 2 ) +1 ) 16π 2 Evaluation of Passarino Veltman functions: In[3]:= PaXEvaluate[B0[SPD[p,p],0,m^2]] 1 ( µ 2 ) m 2 log ( m 2 ) ( Out[3]= ε +log m 2 p 2 m 2 ) +log γ+2 πm 2 p 2 m 2 p 2 Physik-Department T30f (TUM) FeynHelpers 18 / 27

19 We can also expand coefficient functions in their parameters (masses or external momenta). To expand B 0 (p 2, 0, m 2 ) around p 2 = m 2 up to first order with PaXEvaluate we first need to assign an arbitrary symbolic value to the scalar product p 2, e.g. pp In[4]:= SPD[p,p]=pp; Then use the option PaXSeries to specify the expansion parameters and activate the option PaXAnalytic In[5]:= PaXEvaluate[B0[SPD[p,p],0,m^2],PaXSeries {{pp,m^2,1}},paxanalytic True] Out[5]= 3 m2 pp (3 m 2 pp) ( log ( µ 2 ) ) +γ 2+log(π) m 2 2m 2 2εm 2 Get only in the UV-part of this series: PaXEvaluate with PaXEvaluateUV In[6]:= PaXEvaluateUV[B0[SPD[p,p],0,m^2],PaXSeries {{pp,m^2,1}},paxanalytic True] 1 Out[6]= ε UV The IR-part is equally easy In[7]:= PaXEvaluateIR[B0[SPD[p,p],0,m^2],PaXSeries {{pp,m^2,1}},paxanalytic True] Out[7]= m2 pp 2m 2 ε IR Full result with the explicit distinction between UV and IR singularities In[8]:= PaXEvaluateUVIRSplit[B0[SPD[p,p],0,m^2],PaXSeries {{pp,m^2,1}},paxanalytic True] Out[8]= m2 pp (3 m 2 pp) ( log ( µ 2 ) ) +γ 2+log(π) m m 2 ε IR 2 m 2 ε UV Physik-Department T30f (TUM) FeynHelpers 19 / 27

20 Interface to FIRE Main function: FIREBurn Reduces scalar multi-loop integrals to simpler ones using IBP-techniques. Takes three arguments (plus options): input expression, list of loop momenta and the list of external momenta. Automatically adds propagators to integrals with incomplete bases of propagators Automatically detects integrals with linearly dependent propagators Current limitations No recognition of integral families Each loop integral is evaluated separately Hence, rather inefficient... Physik-Department T30f (TUM) FeynHelpers 20 / 27

21 IBP-reduce the 1-loop integral In[9]:= Out[9]= d D l [l 2 ] 2 [(l p) 2 m 2 ] 2 FIREBurn[FAD[{l,0,2},{l p,m,2}],{l},{p}] (D 2)(2 D m 2 9 m 2 pp) 2m 2 (m 2 pp) 3 ((l p) 2 m 2 ) (D 3)(D m2 +D pp 4m 2 6 pp) (m 2 pp) 3 l 2.((l p) 2 m 2 ) No dependence on external momenta supply an empty list for the third argument. For d D q 1 d D q 2 d D q 3 example, for [q 2 1 m2 ] 2 [(q 1 + q 3 ) 2 m 2 ][(q 2 q 3 ) 2 ][q 2 2 ]2 In[10]:= Out[10]= FIREBurn[FAD[{q1,m,2},{q1+q3,m},{q2 q3},{q2,0,2}],{q1,q2,q3},{}] (D 3)(3 D 10)(3 D 8) 16(2 D 7)m 4 (q1 2 m 2 ).q2 2.(q2 q3) 2.((q1+q3) 2 m 2 ) Physik-Department T30f (TUM) FeynHelpers 21 / 27

22 My favourite example: Calculation of the QCD on-shell vertex for QCD/NRQCD matching [Manohar, 1997] Physik-Department T30f (TUM) FeynHelpers 22 / 27

23 Reproducing results of Manohar QCD side of the matching: The on-shell vertex function is evaluated using background field formalism [Abbott, 1981, Abbott, 1982] and expanded up to the first order in the relative momentum squared. The abelian and non-abelian diagrams can be parametrized as µ ( ) = igt a ū(p 2) F (V) 1 (q 2 )γ µ + if (V) 2 (q 2 ) σµν q ν u(p 1), (1) 2m µ ( ) = igt a ū(p 2) F (g) 1 (q 2 )γ µ + if (g) 2 (q 2 ) σµν q ν u(p 1), (2) 2m where q p 2 = p 1. Our goal is to compute the form-factors F (V) 1/2 (q2 ) and F (g) 1/2 (q2 ) expanded up to O(q 2 /m 2 ). Physik-Department T30f (TUM) FeynHelpers 23 / 27

24 Not so simple to do with software It is not a total cross-section/decay rate, so fully automatic tools are not useful. Need to expand Passarino Veltman integrals in the relative momentum. Distinguish between UV and IR singularities in DR using different regulators ɛ UV and ɛ IR. Since this is a matching, we want analytic results. With FEYNHELPERS this computation is straight-forward. We use the abbreviation 1/ɛ γ E + log(4π) and use D = 4 2ɛ To compare to the literature we need to switch to D = 4 ɛ via 1/ɛ 2/ɛ and eliminate γ E and log(4π) by substituting µ 2 with µ 2 e γ E (following the conventions 4π of Manohar). Physik-Department T30f (TUM) FeynHelpers 24 / 27

25 Physik-Department T30f (TUM) FeynHelpers 25 / 27

26 Another example: photon and electron self-energies (with full gauge dependence) in massless QED at 2-loops. Requires evaluation of six 2-loop diagrams + + i/pσ 2V (p 2 ), µ ν + µ ν + µ ν i(p 2 g µν p µ p ν )Π 2 (p 2 ), Final results contain only two master integrals Need to use FCMultiLoopTID instead of TID As expected, the vacuum polarization amplitude is gauge invariant, while the electron self-energy depends on the gauge parameter ξ. These results precisely agree with the literature, e.g. Eq and Eq from [Grozin, 2005]. Physik-Department T30f (TUM) FeynHelpers 26 / 27

27 With FeynHelpers many types of calculations that were difficult or hardly feasible with FeynCalc previously become very simple. Goals for future development: improve the integration with Package-X and FIRE but also to add new interfaces to interesting and useful HEP tools. FeynHelpers comes with many examples. Highlight: 1-loop QED renormalization in MS, MS and on-shell schemes with full gauge dependence (also useful for teaching). Physik-Department T30f (TUM) FeynHelpers 27 / 27

28 Backup Abbott, L. (1981). The background field method beyond one loop. Nucl. Phys. B, 185, Abbott, L. (1982). Introduction to the Background Field Method. Acta Phys. Polon., B13, 33. Anastasiou, C. & Lazopoulos, A. (2004). Automatic Integral Reduction for Higher Order Perturbative Calculations. JHEP, 0407, 046. Belanger, G., Boudjema, F., Fujimoto, J., Ishikawa, T., Kaneko, T., Kato, K., & Shimizu, Y. (2006). GRACE at ONE-LOOP: Automatic calculation of 1-loop diagrams in the electroweak theory with gauge parameter independence checks. Phys. Rept., 430, Bodwin, G. T., Braaten, E., & Lepage, G. P. (1995). Rigorous QCD Analysis of Inclusive Annihilation and Production of Heavy Quarkonium. Phys. Rev. D, 51, Physik-Department T30f (TUM) FeynHelpers 27 / 27

29 Backup Borowka, S., Heinrich, G., Jahn, S., Jones, S. P., Kerner, M., Schlenk, J., & Zirke, T. pysecdec: a toolbox for the numerical evaluation of multi-scale integrals. Brambilla, N., Pineda, A., Soto, J., & Vairo, A. (2000). Potential NRQCD: an effective theory for heavy quarkonium. Nucl. Phys. B, 566, 275. Carrazza, S., Ellis, R. K., & Zanderighi, G. (2016). QCDLoop: a comprehensive framework for one-loop scalar integrals. Comput. Phys. Commun., 209, Caswell, W. & Lepage, G. (1986). Effective lagrangians for bound state problems in QED, QCD, and other field theories. Phys. Lett. B, 167(4), Chetyrkin, K. & Tkachov, F. (1981). Integration by parts: The algorithm to calculate β-functions in 4 loops. Nucl. Phys. B, 192(1), Christensen, N. D. & Duhr, C. (2008). FeynRules - Feynman rules made easy. Comput. Phys. Commun., 180, Physik-Department T30f (TUM) FeynHelpers 27 / 27

30 Backup Cullen, G., Guillet, J. P., Heinrich, G., Kleinschmidt, T., Pilon, E., Reiter, T., & Rodgers, M. (2011). Golem95C: A library for one-loop integrals with complex masses. Comput. Phys. Commun., 182, Cullen, G., van Deurzen, H., Greiner, N., Heinrich, G., Luisoni, G., Mastrolia, P., Mirabella, E., Ossola, G., Peraro, T., Schlenk, J., von Soden-Fraunhofen, J. F., & Tramontano, F. (2014). GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond. Eur. Phys. J. C, 74, 8, Cyrol, A. K., Mitter, M., & Strodthoff, N. (2016). FormTracer - A Mathematica Tracing Package Using FORM. Denner, A., Dittmaier, S., & Hofer, L. (2017). Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations. Comput. Phys. Commun., 212, Feng, F. (2012). $Apart: A Generalized Mathematica Apart Function. Comput. Phys. Commun., 183, Physik-Department T30f (TUM) FeynHelpers 27 / 27

31 Backup Fleischer, J. & Riemann, T. (2011). A complete algebraic reduction of one-loop tensor Feynman integrals. Phys. Rev. D, 83, Grozin, A. (2005). Lectures on QED and QCD. In 3rd Dubna International Advanced School of Theoretical Physics Dubna, Russia, January 29-February 6, 2005 (pp ). Hahn, T. & Perez-Victoria, M. (1999). Automatized One-Loop Calculations in 4 and D dimensions. Comput. Phys. Commun., 118, Jamin, M. & Lautenbacher, M. E. (1993). TRACER version 1.1. Comput. Phys. Commun., 74(2), Lee, R. N. (2012). Presenting LiteRed: a tool for the Loop InTEgrals REDuction. Manohar, A. (1997). The HQET/NRQCD Lagrangian to order α/m 3. Physik-Department T30f (TUM) FeynHelpers 27 / 27

32 Backup Phys. Rev. D, 56, Mertig, R., Böhm, M., & Denner, A. (1991). Feyn Calc - Computer-algebraic calculation of Feynman amplitudes. Comput. Phys. Commun., 64(3), Nogueira, P. (1993). Automatic Feynman graph generation. J. Comput. Phys., 105, Passarino, G. & Veltman, M. (1979). One Loop Corrections for e + e Annihilation Into µ + µ in the Weinberg Model. Nucl. Phys., B160, 151. Patel, H. H. (2015). Package-X: A Mathematica package for the analytic calculation of one-loop integrals. Comput. Phys. Commun., 197, Pineda, A. & Soto, J. (1998a). Effective Field Theory for Ultrasoft Momenta in NRQCD and NRQED. Nucl.Phys.Proc.Suppl., 64, Pineda, A. & Soto, J. (1998b). Physik-Department T30f (TUM) FeynHelpers 27 / 27

33 Backup The Lamb Shift in Dimensional Regularization. Phys. Lett. B, 420, Ruijl, B., Ueda, T., & Vermaseren, J. A. M. (2017). Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams. Shtabovenko, V. (2016). FeynHelpers: Connecting FeynCalc to FIRE and Package-X. Shtabovenko, V., Mertig, R., & Orellana, F. (2016). New Developments in FeynCalc 9.0. Comput. Phys. Commun., 207, Smirnov, A. V. (2015). FIRE5: a C++ implementation of Feynman Integral REduction. Comput. Phys. Commun., 189, Smirnov, A. V. & Smirnov, V. A. (2013). FIRE4, LiteRed and accompanying tools to solve integration by parts relations. Comput. Phys. Commun., 184(12), Studerus, C. (2009). Physik-Department T30f (TUM) FeynHelpers 27 / 27

34 Backup Reduze - Feynman Integral Reduction in C++. Comput. Phys. Commun., 181, Tentyukov, M. & Fleischer, J. (2000). A Feynman Diagram Analyser DIANA. Comput. Phys. Commun., 132, van Hameren, A. (2011). OneLOop: for the evaluation of one-loop scalar functions. Comput. Phys. Commun., 182, van Oldenborgh, G. (1991). FF - a package to evaluate one-loop Feynman diagrams. Comput. Phys. Commun., 66(1), Wang, J.-X. (2004). Progress in FDC project. Nucl. Instrum. Meth. A, 534(1-2), Wiebusch, M. (2014). HEPMath: A Mathematica Package for Semi-Automatic Computations in High Energy Physics. Comput. Phys. Commun., 195, Physik-Department T30f (TUM) FeynHelpers 27 / 27